In[]:=
G[{ϵ_,δ_},{β_,γ_},{βs_,γs_},x_,y_,c1_,c2_,W_]:=Module{n,m1,ν1,m2,ν2,s,p,q,t},​​n=Length[ϵ[[1]]];s=Length[δ[[2]]];p=Length[ϵ[[2]]];m1=Length[β[[1]]];m2=Length[βs[[1]]];ν1=Length[γ[[1]]];ν2=Length[γs[[1]]];q=Length[β[[2]]];t=Length[γ[[2]]];​​(*Thefinalvalueiscalculated*)Print[" Numerical Integration :"];value=-
1
4
2
π
NIntegrate
n
∏
j=1
Gamma[1-ϵ[[1,j]]+(ξ+η)]
p
∏
j=1
Gamma[ϵ[[2,j]]-(ξ+η)]
s
∏
j=1
Gamma[δ[[2,j]]+(ξ+η)]
m1
∏
j=1
Gamma[β[[1,j]]-ξ]
ν1
∏
j=1
Gamma[γ[[1,j]]+ξ]
q
∏
j=1
Gamma[1-β[[2,j]]+ξ]
t
∏
j=1
Gamma[1-γ[[2,j]]-ξ]
m2
∏
j=1
Gamma[βs[[1,j]]-η]
ν2
∏
j=1
Gamma[γs[[1,j]]+η]
q
∏
j=1
Gamma[1-βs[[2,j]]+η]
t
∏
j=1
Gamma[1-γs[[2,j]]-η]
ξ
x
η
y
,{ξ,c1-IW,c1+IW},{η,c2-IW,c2+IW};(*Thevalueisreturned*)Return[value];;
The comparison of (3) in “AN EXTENSION OF MEIJER’S G-FUNCTION” and chapter 3. “Certain Particular Cases of ...” (v) shows an error that has been corrected here.
In[]:=
f[ϵ_,β_,γ_,δ_,βs_,γs_,x_,y_]:=
β
x
βs
y
Gamma[1-ϵ+β+βs]Gamma[γ+β]Gamma[γs+βs]
Gamma[δ+β+βs]
AppellF1[1-ϵ+β+βs,γ+β,γs+βs,δ+β+βs,-x,-y]//N
In[]:=
f
1
3
,
1
5
,
1
3
,
3
5
,
4
5
,
1
5
,
1
2
,2
Out[]=
0.649877
In[]:=
G
1
3
,{},{},
3
5
,
1
5
,{},
1
3
,{},
4
5
,{},
1
5
,{},
1
2
,2,
1
2
,
1
4
,∞
Numerical Integration :
Out[]=
0.649877+4.75191×
-11
10
